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MOCK EXAMINATION

AMC 10 American Mathematics Contest 10

Test Sample

Make time to take the practice test.

It’s one of the best ways to get ready for the AMC.

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AMC 10 Mock Test Problems

Problem 1

Several three-legged tables and four-legged tables have a total of 23 legs. If there is more than

one table of each type, what is the number of three-legged tables?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Problem 2

A bicycle travels at a constant speed of 9 miles per hour. A car starts 200 miles behind the

bicycle and catches up to the bicycle in 4 hours. What is the average speed of the car in miles per

hour?

(A) 40 (B) 50 (C) 59 (D) 60 (E) 65

Problem 3

Twelve 1 by 1 squares form a rectangle, as shown. What is the total area of the shaded regions?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 10



Problem 4

The values of , , , and are 2, 3, 4, and 5, but not necessarily in that order. What is the

maximum possible value of ∙ + ∙ + ∙ ?

(A) 24 (B) 33 (C) 40 (D) 45 (E) 49

Problem 5

There are 400 students at Einstein High School, where the ratio of boys to girls is 3 : 2. There are

600 students at Edison High School, where the ratio of boys to girls is 2 : 3. When considering

all the students from both schools, what is the ratio of boys to girls?

(A) 2 : 3 (B) 12 : 13 (C) 1 : 1 (D) 6 : 5 (E) 3 : 2

Problem 6

The price of each item at the Gauss Gadget Store has been reduced by 20% from its original

price. A printer has a sale price of $112. What would the same printer sell for if it was on sale for

30% off of its original price?

(A) $78.40 (B) $89.60 (C) $98.00 (D) $100.80 (E) $168.00

Problem 7

How many ordered pairs , of positive integers satisfy + = ?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7



Problem 8

How many different ordered pairs , can be formed using numbers from the set of integers { , , ⋯ , } such that < and + is even?

(A) 105 (B) 210 (C) 315 (D) 420 (E) 900

Problem 9

A sequence consists of 2018 terms. Each term after the first is 1 greater than the previous term.

The sum of the 2018 terms is 5311. What is the sum of all odd-numbered terms in the sequence?

(A) 2155 (B) 2153 (C) 2151 (D) 2149 (E) 2147

Problem 10

To shovel all of the snow on his driveway, Alex needs 12 hours. Individually, Bob needs 8 hours

to shovel all of Alex's snow, Carl needs 6 hours to shovel all of Alex's snow, and Allison needs 4

hours to shovel all of Alex's snow. If Alex, Bob, Carl, and Dick all work together, how many

minutes do they need to shovel all of Alex's snow?

(A) 120 (B) 108 (C) 96 (D) 90 (E) 84

Problem 11

If and are positive integers with + = , then the largest possible value of is

(A) 238 (B) 240 (C) 242 (D) 248 (E) 255

Problem 12

A square has side length 5. In how many different locations can point be placed so that the

distances from to the four sides of the square are 1, 2, 3, and 4?



(A) 0 (B) 4 (C) 8 (D) 12 (E) 16

Problem 13

Michael wants to fill his swimming pool using two hoses, each of which sprays water at a

constant rate. Hose X fills the pool in hours when used by itself, where is a positive integer.

Hose Y fills the pool in hours when used by itself, where is a positive integer. When used

together, Hose X and Hose Y fill the pool in 10 hours. How many different possible values are

there for ?

(A) 3 (B) 6 (C) 7 (D) 8 (E) 9

Problem 14

Two circles each with radius 2 overlap so that each contains exactly 25% of the other's

circumference, as shown. The area of the shaded region is

(A) 𝜋 − (B) 𝜋 − (C) 𝜋 (D) 𝜋 − (E) 𝜋 −

Problem 15

Each of the five boxes is to contain a number, as shown below. Each number in a shaded box

must be the average of the number in the box to the left of it and the number in the box to the

right of it. What is the value of ?



(A) 28 (B) 30 (C) 31 (D) 32 (E) 34

Problem 16

A rectangular piece of paper measures 17 inches by 8 inches. It is folded so that a right angle is

formed between the two segments of the original bottom edge, as shown. What is the area in

square inches of the new figure?

(A) 64 (B) 72 (C) 81 (D) 104 (E) 168

Problem 17

In a sequence of 10 terms, the first term is 1, the second term is , and each term after the second

is the sum of the previous two terms. For example, if = , the sequence would be , , , , , , , , , . For some values of , the number 463 appears in the

sequence. If is a positive integer, what is the number of all the possible values of for which

463 appears in the sequence?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8



Problem 18

Michelle assigns every letter of the alphabet a different positive integer value. She finds the

value of a word by multiplying the values of its letters together. For example, if 𝐷 has a value of

5, and 𝐴 has a value of 11, then the word 𝐷𝐴𝐷 has a value of × × = . The table

shows the value of some words. What is the value of the word MATH?

Word Value

TOTE 18

TEAM 168

MOM 49

HOME 70

MATH ?

(A) 19 (B) 190 (C) 420 (D) 84 (E) 840

Problem 19

Rhombus is inscribed in rectangle , as shown. Segments and are parallel to . Segments and are parallel to . If = , = , and = , what is the

perimeter of rectangle ?



(A) 48 (B) 58 (C) 84 (D) 96 (E) 108

Problem 20

Let , , , ⋯ , 7, be a sequence defined by

𝑘+ = + √ ∙ 𝑘√ − 𝑘

with = . What is the value of 7?

(A) (B) (C) √ (D) (E) √

Problem 21

The product of consecutive four digit positive integers is divisible by 2010. What is the least

possible value of ?

(A) 5 (B) 12 (C) 10 (D) 6 (E) 7

Problem 22

Six baseball teams are competing in a tournament in New York City. Every team is to play three

games, each against a different team. (Note that not every pair of teams plays a game together.)

Bob is in charge of pairing up the teams to create a schedule of games that will be played.

Ignoring the order and times of the games, how many different schedules are possible?

(A) 60 (B) 70 (C) 80 (D) 90 (E) 100



Problem 23

A rectangular piece of paper, , has = and = . The piece of paper is glued at

on the surface of a large cube so that and are at vertices of the cube. (Note that ∆ and ∆ lie at on the front and top faces of the cube, respectively.) The shortest distance from to , as measured through the cube, is

(A) √ (B) √ (C) √ (D)√ (E)

Problem 24

Alan has a number of gold bars, all of different weights. He gives the 24 lightest bars, which

weigh 45% of the total weight, to Bob. He gives the 13 heaviest bars, which weigh 26% of the

total weight, to Carl. She gives the rest of the bars to Dale. How many bars did Dale receive?

(A) 19 (B) 18 (C) 17 (D) 16 (E) 15

Problem 25

A coin that is 8 units in diameter is tossed onto a 5 by 5 grid of squares each having side length

10 units. A coin is in a winning position if no part of it touches or crosses a grid line, otherwise it



is in a losing position. Given that the coin lands in a random position so that no part of it is off

the grid, what is the probability that it is in a winning position?

(A) 9 (B) 7 (C) (D) (E) 𝜋

These problems are copyright Ivy League Education Center

(https://ivyleaguecenter.wordpress.com/)

https://ivyleaguecenter.wordpress.com/



Answer Key

1. E

2. C

3. E

4. D

5. B

6. C

7. A

8. B

9. C

10. C

11. B

12. C

13. E

14. B

15. D

16. D

17. A

18. C

19. D

20. A

21. A

22. B

23. B

24. E

25. D



Ivy League Education Center

(https://ivyleaguecenter.wordpress.com/)

https://ivyleaguecenter.wordpress.com/

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